SOAP FILMS

Knowing the possible shapes that a soap film can take once its contours are already pre-defined (by metals) is what is called, in Mathematics, the Plateau problem.

When the film does not surround the air forming closed chambers, it has the property of minimizing area. This means that any nearby shape that has the same outline will necessarily have more area.

These shapes are therefore called minimal surfaces. Note, especially when the contours are edges of a polyhedron, that there are also internal edges created by the meeting of skins, which are sometimes even in planar form. This creates beautiful figures, sometimes more than one for the same outline.

The study of minimal surfaces is included in what is modernly called differential geometry.

Some examples of minimal surfaces.

This Wikipedia page brings an overview of the Plateau Problem, giving its exact definition and telling a little of its history, that is, what led to its study. In addition, it brings an explanation of what happens in higher dimensions (surfaces in k dimensions in n-dimensional spaces) and also an application in Physics!                                                                                       

As stated, the idea of the Plateau Problem is to find the minimal surfaces, which are described in this Wikipedia page. In it, its definition is given and several examples where it can be applied, in addition to explaining the generating equation of its study. Finally, it makes some generalizations and connections with other mathematical fields.

Finally, we saw that the area of Mathematics that studies the Plateau Problem today is differential geometry. In this Wikipedia page, a little is known about its history and the different branches in which it opens. It also brings some of its applications in other fields, such as physics, chemistry and even economics!